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e5a97c6 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 | import Mathlib
import OAI.Combinatorics.MatchingCount.Machines.TreeReach
import OAI.Combinatorics.MatchingCount.Machines.TreeStateTable
namespace OAI
namespace MatchingFPRAS.ExecutableLadder
open TreeTyped Polynomial
variable {I E : Type} [Codec I] [Codec E] [DecidableEq I] [Inhabited I] [LinearOrder E]
abbrev StateEnvelope (I E : Type) := List I×Finset E
def normalizeState (x : StateEnvelope I E×StateTable I (Finset E)) : StateTable I (Finset E) :=
x.1.1.map (fun i=>(i,lookup ∅ (x.2,i)∩x.1.2))
def normalizeStateRealizer (Eq : Realizer (fun x : I×I=>decide (x.1=x.2)))
(EqE : Realizer (fun x : E×E=>decide (x.1=x.2))) : Realizer (normalizeState (I:=I) (E:=E)) :=
(composition (pair identity first.fst) (mapWith (pair second
(composition (pair (composition (pair first.snd second) (lookupRealizer ∅ Eq)) first.fst.snd)
(finsetInter EqE))))).congr (fun _=>rfl)
lemma polynomial_normalizeState {Eq : Realizer (fun x : I×I=>decide (x.1=x.2))}
{EqE : Realizer (fun x : E×E=>decide (x.1=x.2))}
(hEq : PolynomialTime Eq) (hE : PolynomialTime EqE) : PolynomialTime (normalizeStateRealizer Eq EqE) := by
unfold normalizeStateRealizer Realizer.fst Realizer.snd
apply polynomial_congr
have := polynomial_lookup (∅ : Finset E) hEq
tree_cost
omit [Inhabited I] in
lemma normalizeState_weight (keys : List I) (univ : Finset E) (s : StateTable I (Finset E)) :
weight (normalizeState ((keys,univ),s))≤(weight keys+1)*(weight univ+3) := by
induction keys with
| nil=>simp only [normalizeState,List.map_nil,weight_nil]; omega
| cons i keys ih=>
have hu : weight (lookup ∅ (s,i)∩univ)≤weight univ := weight_finset_subset Finset.inter_subset_right
change weight ((i,lookup ∅ (s,i)∩univ)::normalizeState ((keys,univ),s))≤_
rw [weight_cons,weight_prod,weight_cons]
nlinarith
omit [Codec I] [Codec E] [Inhabited I] in
lemma normalizeState_rep {Ω : Type} (keys : List I) (univ : Finset E) (e : Ω→Finset E)
(he : ∀ a,e a⊆univ) (s : I→Ω) :
normalizeState ((keys,univ),stateRep keys (e ∘ s))=stateRep keys (e ∘ s) := by
unfold normalizeState stateRep
apply List.map_congr_left
intro i hi
change (i,lookup ∅ (stateRep keys (e ∘ s),i)∩univ)=(i,e (s i))
rw [lookup_stateRep ∅ keys (e ∘ s) i hi]
simp only [Function.comp_apply,Finset.inter_eq_left.mpr (he (s i))]
end MatchingFPRAS.ExecutableLadder
end OAI
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